BIFURCATION ANALYSIS OF A KALDOR-KALECKI MODEL OF BUSINESS CYCLE WITH TIME DELAY
Liancheng Wang 1 and Xiaoqin P. Wu 2
1 Department of Mathematics & Statistics Kennesaw State University
Kennesaw, GA 30144, USA e-mail: lwang5@kennesaw.edu
2 Department of Mathematics, Computer & Information Sciences Mississippi Valley State University
Itta Bena, MS 38941, USA e-mail: xpaul wu@yahoo.com
Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday Abstract
In this paper, we investigate a Kaldor-Kalecki model of business cycle with delay in both the gross product and the capital stock. Stability analysis for the equilibrium point is carried out. We show that Hopf bifurcation occurs and pe- riodic solutions emerge as the delay crosses some critical values. By deriving the normal forms for the system, the direction of the Hopf bifurcation and the stabil- ity of the bifurcating periodic solutions are established. Examples are presented to confirm our results.
Key words and phrases: Kaldor-Kalecki model of business cycle, Hopf bifurcation, periodic solutions, stability.
AMS (MOS) subject classifications: 34K18
1 Introduction
In this paper, we study the Kaldor-Kalecki model of business cycle with delay of the following form:
( dY(t)
dt =α[I(Y(t), K(t))−S(Y(t), K(t))],
dK(t)
dt =I(Y(t−τ), K(t−τ))−qK(t), (1) whereY is the gross product,K is the capital stock, α >0 is the adjustment coefficient in the goods market, q ∈ (0,1) is the depreciation rate of capital stock, I(Y, K) and S(Y, K) are investment and saving functions, and τ ≥ 0 is a time lag representing delay for the investment due to the past investment decision.
A business model in this line was first proposed by Kalecki [11], in which the idea of a delay of the implementation of a business decision was introduced. Later on, Kalecki [12] and Kaldor [10] proposed and studied business models using ordinary differential equations and nonlinear investment and saving functions. They showed that periodic solutions exist under the assumption of nonlinearity. Similar models were also analyzed by several authors and the existence of limit cycles were established due to the nonlinearity, see [4, 7, 23]. Krawiec and Szydlowski [14, 15, 16] combined the two basic models of Kaldor’s and Kalecki’s and proposed the following Kaldor-Kalecki model of business cycle:
( dY
(t)
dt =α[I(Y(t), K(t))−S(Y(t), K(t))],
dK(t)
dt =I(Y(t−τ), K(t))−qK(t).
This model has been studied intensively since its introduction, see [17, 19, 20, 21, 22, 24]. It is argued that a more reasonable model should include delays in both the gross product and capital stock, because the change in the capital stock is also caused by the past investment decisions [17]. Adding a delay to capital stock K leads to System (1).
As in [14], also see [1, 2, 22], using the following saving and investment functions S and I, respectively,
S(Y, K) = γY, I(Y, K) =I(Y)−βK
where β >0 andγ ∈(0,1) are constants, System (1) becomes the following system:
( dY(t)
dt =α[I(Y(t))−βK(t)−γY(t)],
dK(t)
dt =I(Y(t−τ))−βK(t−τ)−qK(t). (2) Kaddar and Talibi Alaoui [9] studied System (2). They gave a condition for the charac- teristic equation of the linearized system to have a pair of purely imaginary roots and showed that the Hopf bifurcation may occur as the delay τ passes some critical values.
However, they did not give the stability of the periodic solution and the direction of the Hopf bifurcation.
In this paper, we first give a more detailed discussion of the distribution of the eigenvalues of the linearized system of (2). So local stability of the equilibrium point is established. Conditions are found under which the Hopf bifurcation occurs and peri- odic solutions emerge as the delay crosses some critical values. By deriving the normal forms for System (2) using the normal form theory developed by Faria and Magalh˜aes [5, 6], the direction of the Hopf bifurcation and the stability of the bifurcating peri- odic solutions are established. Finally, some examples are presented to illustrate our theoretical results.
2 Distribution of Eigenvalues
Throughout the rest of this paper, we assume that I(s) is a nonlinear function, C3, and that System (2) has an isolated equilibrium point (Y∗, K∗). Let I∗ = I(Y∗), u1 = Y − Y∗, u2 = K −K∗, and i(s) = I(s +Y∗)−I∗. Then System (2) can be transformed into
( du1(t)
dt =α[i(u1(t))−βu2(t)−γu1(t)],
du2(t)
dt =i(u1(t−τ))−βu2(t−τ)−qu2(t). (3)
Let the Taylor expansion of iat 0 be
i(u) = ku+i(2)u2+i(3)u3+O(|u|4) where
k=i′(0) =I′(Y∗), i(2) = 1
2i′′(0) = 1
2I′′(Y∗), i(3) = 1
3!i′′′(0) = 1
3!I′′′(Y∗).
The linear part of System (3) at (0,0) is ( du1(t)
dt =α[(k−γ)u1(t)−βu2(t)],
du2(t)
dt =ku1(t−τ)−βu2(t−τ)−qu2(t), (4)
and its corresponding characteristic equation is
λ2+ [q−α(k−γ)]λ−αq(k−γ) + (βλ+αβγ)e−λτ = 0. (5) For τ = 0, Equation (5) becomes
λ2+ [q+β−α(k−γ)]λ−αq(k−γ) +αβγ = 0. (6) Define
k1 = βγ
q +γ, k2 = q+β α +γ,
and for the rest of the paper, we always assumek1 ≤k2.For the case thatk1 > k2, the discussion can be carried out similarly.
Theorem 2.1. Let τ = 0. Ifk < k1, all roots of Equation (6) have negative real parts, and hence(Y∗, K∗)is asymptotically stable. If k > k1, Equation (6) has a positive root and a negative root, and hence (Y∗, K∗) is unstable.
Now assume τ > 0. Let ωi (ω > 0) be a purely imaginary root of Equation (5).
After plugging it into Equation (5) and separating the real and imaginary parts, we have
ω2+αq(k−γ) = αβγcos(ωτ) +βωsin(ωτ),
[q−α(k−γ)]ω = αβγsin(ωτ)−βωcos(ωτ). (7) Adding squares of two equations yields
ω4+ [q2−β2+α2(k−γ)2]ω2+α2q2(k−γ)2−α2β2γ2 = 0. (8) Let
A = q2−β2+α2(k−γ)2, B = α2q2(k−γ)2−α2β2γ2.
If A≥0 and B ≥ 0, Equation (8) has no positive roots. If B <0,Equation (8) has a unique positive root
ω+ = s
−A+√
A2−4B
2 .
If A <0, B >0,and A2−4B >0, Equation (8) has two positive roots ω± =
s
−A±√
A2−4B
2 .
Solving Equation (7) for sin(ωτ) and cos(ωτ) yields
sin(ωτ) = ω3+ [αqk−α2γ(k−γ)]ω α2βγ2+βω2 , cos(ωτ) = α2qγ(k−γ) + (αk−q)ω2
α2βγ2+βω2 . Define
l1± = ω3±+ [αqk−α2γ(k−γ)]ω±
α2βγ2+βω±2 , l2± = α2qγ(k−γ) + (αk−q)ω2±
α2βγ2+βω±2 . We, thus, have the following result.
Lemma 2.1. Let ω± and l±i (i= 1,2) be defined above.
(i) If B < 0, then there exists a sequence of positive numbers {τj+}∞j=0 such that τ0+ < τ1+ < τ2+ < · · · < τj+ < · · ·, and Equation (5) has a pair of purely imaginary roots ±iω+ when τ =τj+.
(ii) If A < 0, B > 0, and A2 −4B > 0, then there exist two sequences of positive numbers {τj+}∞j=0 and {τj−}∞j=0 such that τ0+ < τ1+ < τ2+ <· · ·< τj+<· · · , τ0−<
τ1− < τ2− < · · · < τj− < · · · , and Equation (5) has a pair of purely imaginary roots ±iω± whenτ =τj±.
Here τj± (j = 0,1,2,· · ·) are defined below τj± = 1
ω±
arccosl±2 + 2jπ, if l±1 >0, 2π−arccosl±2 + 2jπ, if l±1 <0.
Define λ(τ) =σ(τ) +iω(τ) to be the root of Equation (5) such thatσ(τj±) = 0 and ω(τj±) =ω±, respectively.
Lemma 2.2. Let σ(τ) and τj± be defined above. Then σ′(τj+)>0, σ′(τj−)<0.
Proof. Differentiate Equation (5) with respect to τ yields dλ
dτ −1
= [2λ+q−α(k−γ)]eλτ +β λβ(λ+αγ) − τ
λ and a calculation gives
Re dλ
dτ −1
τ=τj± = 2ω±2 +α2(k−γ)2+q2−β2
β2(α2γ2 +ω±2) = 2ω±2 +A β2(α2γ2+ω2±) which gives
Re dλ
dτ −1
τ=τj± = ±√
A2−4B β2(α2γ2+ω2±), completing the proof.
To discuss the distribution of the roots of Equation (5), we will need the following lemma due to Ruan and Wei [18].
Lemma 2.3. Consider the exponential polynomial P(λ, e−λτ) =p(λ) +q(λ)e−λτ
where p, q are real polynomials such that deg(q)<deg(p) and τ ≥ 0. As τ varies, the total number of zeros of P(λ, e−λτ) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
Now we turn our attention to the relationship between A, B and our system pa- rameters. We look at the following two cases.
Case I. β≤q. In this case, A≥0.
1. B ≥0⇐⇒k ≥k1 ork ≤ −βγ/q+γ; 2. B <0⇐⇒ |k−γ|< βγ/q.
Case II.β > q. In this case,
1. A ≥0, B ≥ 0⇐⇒k ≥max{p
β2−q2/α+γ, k1} or k ≤ min{−p
β2−q2/α+ γ,−βγ/q+γ};
2. B <0⇐⇒ |k−γ|< βγ/q;
3. A <0, B >0⇐⇒βγ/q <|k−γ|<p
β2−q2/α.
The discussions above, Theorem 2.1 and Lemmas 2.1, 2.2 and 2.3 imply the following Lemma 2.4.
Lemma 2.4. Assume β ≤q. Let τj+ be defined in Lemma 2.1. Then we have
(i) if B ≥ 0, then all roots of Equation (5) have negative real parts when k <
−βγ/q+γ and Equation (5) has roots with negative real parts and roots with positive real parts when k > k1;
(ii) if B <0, or |k−γ|< βγ/q, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ0+); Equation (5) has a pair of purely imaginary roots ±iω+ and all other roots have negative real parts whenτ =τ0+; it has2(j+1)roots with positive real parts and all other roots have negative real parts when τ ∈ (τj+, τj+1+ ), j = 0,1,2,· · · .
Lemma 2.5. Assume β > q. Let τj± be defined in Lemma 2.1. Then we have
(i) if A ≥ 0, B ≥ 0, then all roots of Equation (5) have negative real parts when k <min{−p
β2−q2/α+γ,−βγ/q+γ},and Equation (5) has roots with negative real parts and roots with positive real parts when k >max{p
β2−q2/α+γ, k1}; (ii) if B < 0, or if |k−γ|< βγ/q, all roots of Equation (5) have negative real parts for allτ ∈ [0, τ0+);Equation (5) has a pair of purely imaginary roots ±iω+ and all other roots have negative real parts whenτ =τ0+; it has2(j+1)roots with positive real parts and all other roots have negative real parts when τ ∈ (τj+, τj+1+ ), j = 0,1,2,· · · .
(iii) ifA <0, B >0andA2−4B >0,then we haveβγ/q <|k−γ|<p
β2−q2/αand A2−4B >0. Assume thatA2−4B >0. If −p
β2 −q2/α+γ < k <−βγ/q+γ, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ0+), Equation (5) has roots with positive real parts when τ ∈ (τ0+, τm−) where m is the smallest positive integer such that τm− > τ0+, it has a pair of purely imaginary roots±iω+
and all other roots have negative real parts when τ = τ0+. if τm− < τ1+, Equation
(5) has two roots with positive real parts and all other roots have negative real parts when τ ∈ (τ0+, τm−) and all roots of Equation (5) have negative real parts when τ ∈ (τm−, τ1+). If k1 < k < p
β2−q2/α+γ, Equation (5) has roots with negative real parts and roots with positive real parts.
The following Hopf bifurcation theorems follow immediately.
Theorem 2.2. Assume β ≤q. Let τj+ be defined in Lemma 2.1. Then we have
(i) the equilibrium point (Y∗, K∗) is asymptotically stable for all τ ≥ 0 when k <
−βγ/q+γ and it is unstable for all τ ≥0 when k > k1;
(ii) the equilibrium point (Y∗, K∗) is asymptotically stable for all τ ∈ [0, τ0+) and unstable for all τ > τ0+ when |k −γ| < βγ/q. System (2) undergoes a Hopf bifurcation at (Y∗, K∗) whenτ =τj+ for j = 0,1,2,· · · .
Theorem 2.3. Assume β > q. Let τj± be defined in Lemma 2.1. Then we have
(i) the equilibrium point (Y∗, K∗) is asymptotically stable for all τ ≥ 0 when k <
min{−p
β2−q2/α + γ,−βγ/q + γ}, and unstable for all τ ≥ 0 when k >
max{p
β2−q2/α+γ, k1};
(ii) the equilibrium point (Y∗, K∗) is asymptotically stable for all τ ∈ [0, τ0+) and unstable for all τ > τ0+ when |k − γ| < βγ/q. System (2) undergoes a Hopf bifurcation at (Y∗, K∗) whenτ =τj+ for j = 0,1,2,· · · .
(iii) Assume A2−4B >0. The equilibrium point(Y∗, K∗)is asymptotically stable for all τ ∈[0, τ0+) and unstable when τ ∈(τ0+, τm−) where m is defined in Lemma 2.5 when−p
β2−q2/α+γ < k <−βγ/q+γ. System (2) undergoes a Hopf bifurca- tion at(Y∗, K∗)whenτ =τj+ forj = 0,1,2,· · · .Whenk1 < k <p
β2−q2/α+γ, the equilibrium point (Y∗, K∗) is unstable. System (2) undergoes a Hopf bifurca- tion at (Y∗, K∗) when τ =τj± for j = 0,1,2,· · · .
3 Direction and Stability of Hopf Bifurcation
From Section 2, we know that at (Y∗, K∗) the characteristic equation of linearized System (2) has a pair of purely imaginary roots±iω± if τ =τj± for each j under some conditions. Under these conditions, as the delay τ passes the critical values τj±, Hopf bifurcation occurs and periodic solutions emerge. In this section, by deriving a normal form for System (2) using a normal form theory developed by Faria and Magalh˜aes [5, 6], we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.
We first normalize the delay in System (2) by rescaling t→t/τ to get the following system
du1(t)
dt = ατ[(k−γ)u1(t)−βu2(t) +i(2)u21(t) +i(3)u31(t)] +O(|u1|4),
du2(t)
dt = τ[ku1(t−1)−βu2(t−1)−qu2(t) +i(2)u21(t−1) +i(3)u31(t−1)] +O(|u1|4).
(9)
Let τc =τj± and τ = τc +µ. Then µ is the bifurcation parameter for System (9) and System (9) becomes
du1(t)
dt = α(τc+µ)[(k−γ)u1(t)−βτ u2(t) +i(2)u21(t) +i(3)u31(t)]
+O(|u1|4),
du2(t)
dt = (τc+µ)[ku1(t−1)−βu2(t−1)−qu2(t) +i(2)u21(t−1) +i(3)u31(t−1)] +O(|u1|4).
(10)
The linearization of System (10) at (0,0) is ( du
1(t)
dt =ατc[(k−γ)u1(0)−βu2(0)],
du2(t)
dt =τc[ku1(−1)−βu2(−1)−qu2(0)]. (11) Let
η(θ) = Aδ(θ) +Bδ(θ+ 1) where
A=τc
α(k−γ) −αβ
0 −q
, B =τc
0 0 k −β
. Let C=C([−1,0],C2) and define a linear operator Lon C as follows:
Lϕ = Z 0
−1
dη(θ)ϕ(θ), ∀ϕ∈C.
Then System (10) can be transformed into
X(t) =˙ LXt+F(Xt, µ),
where X = (u1, u2)T, Xt=X(t+θ), θ∈[−1,0], and F(Xt, µ) = (F1, F2)T where F1 = α[(k−γ)µu1(0)−βµu2(0) +τci(2)u21(0) +τci(3)u31(0)] + h.o.t.,
F2 = kµu1(−1)−βµu2(−1)−qµu2(0) +τci(2)u31(−1) +τci(3)u21(−1) + h.o.t., where “h.o.t” represents high order terms. Write the Taylor expansion of F as
F(ϕ, µ) = 1
2F2(ϕ, µ) + 1
3!F3(ϕ, µ) + h.o.t..
Take the enlarged space of C
BC ={ϕ: [−1,0]→C2 : ϕ is continuous on [−1,0), ∃ lim
θ→0−ϕ(θ)∈C2}. Then the elements of BC can be expressed asψ =ϕ+X0ν,ϕ ∈C and
X0(θ) =
0, −1≤θ <0, I, θ= 0,
where I is the identity matrix on C and the norm of BC is |ϕ+X0ν| = |ϕ|+|ν|C2. Let C1 = C1([−1,0],C2). Then the infinitesimal generator A : C1 → BC associated with L is given by
Aϕ= ˙ϕ+X0[Lϕ−ϕ(0)] =˙
ϕ,˙ −1≤θ <0, Aϕ(0) +Bϕ(−1), θ = 0, and its adjoint
A∗ψ=
−ψ,˙ 0< s≤1,
ψ(0)A+ψ(1)B, s= 0, for ∀ψ ∈C1∗,
where C1∗ = C1([0,1],C2∗). Let C′ =C([0,1],C2∗) and forϕ ∈C and ψ ∈ C′, define a bilinear inner product between C and C′ by
hψ, ϕi = ψ(0)ϕ(0)− Z 0
−1
Z θ 0
ψ(ξ−θ)dη(θ)ϕ(ξ)dξ
= ψ(0)ϕ(0) + Z 0
−1
ψ(ξ+ 1)Bϕ(ξ)dξ.
From Section 2, we know that ±iτcω0 are eigenvalues of A and A∗, where ω0 = ω+
or ω−. Now we compute eigenvectors of A associated with iτcω0 and eigenvectors of A∗ associated with −iτcω0. Let q(θ) = (ρ, k)Teiτcω0θ be an eigenvector ofA associated with iτcω0. Then Aq(θ) =iτcω0q(θ). It follows from the definition of A that
−α(k−γ)τc +iτcω0 αβτc
−kτce−iτcω0 βτce−iτcω0+qτc+iτcω0
q(0) = 0.
We can obviously choose q(θ) = (ρ, k)Teiτcω0θ where ρ=β+ (q+iω0)eiτcω0. Similarly, we can find an eigenvector p(s) of A∗ associated with −iτcω0
p(s) = 1
D(σ, αβ)eiτcω0s, whereσ =−βeiτcω0 −q+iω0
with D being a constant to be determined such that hp(s), q(θ)¯ i= 1. In fact, since hp(s), q(θ)¯ i= 1
D¯[kαβ(1 + (ρ−β)e−iτcω0) +ρ¯σ]
we have D = kαβ(1 + (¯ρ−β)eiτcω0) + ¯ρσ. Let P be spanned by q,q¯and P∗ by p,p.¯ Then C can be decomposed as
C =P ⊕Q where Q={ϕ ∈C :hψ, ϕi= 0,∀ψ ∈P∗}.
Let Q1 = Q∩C1. Let Φ(θ) = (q(θ),q(θ)) and Ψ(s) =¯ p(s)p(s)¯
. Then ˙Φ = ΦJ and Ψ =˙ −JΨ where J = diag(iτcω0,−iτcω0). Define the projectionπ :BC →P by
π(ϕ+X0ν) = Φ[(Ψ, ϕ) + Ψ(0)ν].
Let u= Φx+y, namely
u1(θ) = eiτcω0θρx1 +e−iτcω0θρx¯ 2+y1(θ), u2(θ) = eiτcω0θkx1+e−iτcω0θkx2+y2(θ).
Then System (10) can be decomposed as
x˙ =Jx+ Ψ(0)F(Φx+y, µ),
˙
y=AQ1y+ (I−π)X0F(Φx+y, µ).
This can be rewritten as
x˙ =Jx+12f21(x, y, µ) + 3!1f31(x, y, µ) + h.o.t.,
˙
y=AQ1y+ 12f22(x, y, µ) + 3!1f32(x, y, µ) + h.o.t., (12) where
fj1(x, y, µ) = Ψ(0)Fj(Φx+y, µ), fj2(x, y, µ) = (I−π)X0Fj(Φx+y, µ).
According to the normal form theory due to Faria and Magalh˜aes [5, 6, 8], on the center manifold, System (12) can be transformed as the following normal form:
˙
x=Jx+1
2g21(x,0, µ) + 1
3!g31(x,0, µ) + h.o.t.
wheregj1(x,0, µ) is a homogeneous polynomial of degreej in (x, µ). LetY be a normed space and j, p∈N. Let
Vjp(Y) =
X
|q|=j
cqxq :q ∈Nq
0, cq ∈Y
with norm|P
|q|=jcqxq|=P
|q|=j|cq|Y. DefineMj to be the operator inVj4(C2×kerπ) with the range in the same space by
Mj(p, h) = (Mj1p, Mj2h),
where (Mj1p)(x, µ) = [J, p(·, µ)](x) = Dxp(x, µ)Jx−Jp(x, µ). It is easy to check that Vj3(C2) = Im(Mj1)⊕Ker(Mj1) and
Ker(Mj1) = {µlxqek: (q,¯λ) =λk, k = 1,2, q∈N2
0,|(q, l)|=j}.
Hence
ker(M21) = Span
µx1
0
, 0
µx2
, ker(M31) = Span
x21x2
0
, µ2x1
0
, 0
x1x22
, 0
µ2x2
. Define
f˜31(x,0, µ) =f31(x,0, µ) + 3
2[(Dxf21)(x,0, µ)U21(x, µ) + (Dyf21)(x,0, µ)U22(x, µ)]
where
U21(x, µ)|µ=0 = (M21)−1ProjIm(M1
2)f21(x,0,0) = (M21)−1f21(x,0,0) and U22(x, µ) is determined by
(M22U22)(x, µ) =f22(x,0, µ).
Then
g21(x,0, µ) = Projker(M1
2)f21(x,0, µ), g13(x,0, µ) = Projker(M1
3)f˜31(x,0, µ).
Let us compute g21(x,0, µ) first. Since 1
2f21(x,0, µ) = a1µx1+a2µx2+a20x21+a11x1x2+a02x22
¯
a2µx1+ ¯a1µx2+ ¯a02x21+ ¯a11x1x2+ ¯a20x22
! ,
where
a1 = −α
D¯[kβ(q+ (β−ρ)e−iτcω0) + ¯σ(k(β−ρ) +γρ)], a2 = −α
D¯[kβ(q+ ¯σ+βeiτcω0)−ρ(kβe¯ iτcω0 + (k−γ)¯σ)], a20 = αρ2τc
D¯ i(2)(e−2iτcω0β+ ¯σ), (13) a11 = 2α|ρ|2τc
D¯ i(2)(β+ ¯σ), a02 = αρ¯2τc
D¯ i(2)(e2iτcω0β+ ¯σ), then
1
2g21(x,0, µ) = 1
2Projker(M1
2)f21(x,0, µ) =
a1µx1
¯ a1µx2
.
Next we compute 3!1g31(x,0, µ) = 3!1Projker(M1
3)f˜31(x,0, µ). Since the term O(µ2|x|) is irrelevant to determine the generic Hopf bifurcation, we have
1
3!g31(x,0, µ) = 1
3!Projker(M1
3)f˜31(x,0, µ) = 1
3!ProjSf˜31(x,0,0) +O(µ2|x|)
= 1
3!ProjSf31(x,0,0) + 1
4ProjS[(Dxf21)(x,0,0)U21(x,0) + (Dyf21)(x,0,0)U22(x,0)] +O(µ2|x|).
where
S= Span
x21x2
0
, 0
x1x22
. Step 1. Compute 3!1Projker(M31)f31(x,0,0). Since
1
3!f31(x,0,0) =
a30x31+a21x21x2 +a12x1x22+a03x32
¯
a03x31+ ¯a12x21x2 + ¯a21x1x22+ ¯a30x32
where
a30 = αρ3τc
D¯ [i(3)(βe−3iτcω0 + ¯σ)], a21 = 3α|ρ|2ρτc
D¯ [i(3)(βe−iτcω0 + ¯σ)], a12 = 3α|ρ|2ρτ¯ c
D¯ [i(3)(βeiτcω0 + ¯σ)], a03 = αρ¯3τc
D¯ [i(3)(βe3iτcω0 + ¯σ)], we have
1
3!Projker(M1
3)f31(x,0,0) =
a21x21x2
¯ a21x1x22
.
Step 2. Compute 12ProjS[Dxf21(x,0,0)U21(x,0)]. The elements of the canonical basis of V22(C2) are
x21 0
,
x1x2
0
, x22
0
, µx1
0
, µx2
0
, µ2
0
, 0
x21
, 0
x1x2
,
0 x22
,
0 µx1
,
0 µx2
,
0 µ2
, whose images under iω1
0M21 are, respectively x21
0
,− x1x2
0
,−3 x22
0
, 0
0
,−2 µx2
0
, µ2
0
, 3
0 x21
,
0 x1x2
,−
0 x22
,2
0 µx1
,
0 0
, 0
µ2
.
Hence
U21(x,0) = 1 iω0
a20x21−a11x1x2− 13a02x22
1
3a¯02x21+ ¯a11x1x2 −¯a20x22
,
and 1
2ProjS[Dxf21(x,0,0)U21(x,0)] =
C1x21x2
C¯1x1x22
where
C1 = 2 iω0
(2|a02|2−3a20a11+ 3|a11|2)
= −iα2|ρ|2ρτc2(i(2))2
12 ¯D|D|2 [6 ¯Dρ¯|β+σ|2−3Dρ(β+ ¯σ)(βe−2iτcω0 + ¯σ) + D¯¯ρ|βe−2iτcω0 +σ|2].
Step 3. Compute 12ProjS[(Dyf21)(x,0,0)U22(x,0)], where U22(x,0) is a second-order homogeneous polynomial in (µ, x1, x2) with coefficients in Q1. Let
h(x)(θ) =U22(x,0) =h20(θ)x21+h11(θ)x1x2+h02(θ)x22. The coefficients hjk = (h1jk, h2jk)T are determined by M22h(x) =f22(x,0,0) or
Dxh(x)Bx−AQ1(h(x)) = (I−π)X0F2(Φx,0) which is equivalent to
h(x)˙ −Dxh(x)Bx= ΦΨ(0)F2(Φx,0), h(x)(0)˙ −Lh(x) =F2(Φx,0),
where ˙h denotes the derivative of h(x)(θ) with respect to θ. Note that F2(Φx,0) =A20x21+A11x1x2 +A02x22
where
A20 = (2i(2)αρ2τc,2i(2)ρ2τce−2iτcω0)T, A11 = (4i(2)α|ρ|2τc,2i(2)α|ρ|2τc)T, A02 = (2i(2)αρ¯2τc,2i(2)ρ¯2τce2iτcω0)T.
Comparing the coefficients of x21, x1x2, x22 of these equations, it is not hard to verify that ¯h02 =h20,¯h11=h11 and that h20, h11 satisfy the following equations
h˙20−2iτcω0h20= ΦΨ(0)A20,
h˙20(0)−Lh20=A20, (14)
and
h˙11 = ΦΨ(0)A11,
h˙11(0)−Lh11 =A11. (15)
Noting that f21(x,0,0) = ΨF2(Φx,0), we deduce 1
2(Dyf21)h(x,0,0) =
ατci(2)
2 ¯D [β(ρe−iτcω0x1+ ¯ρeiτcωox2)h1(−1) + ¯σ(ρx1 + ¯ρx2)h1(0)]
ατci(2)
2D [β(ρe−iτcω0x1+ ¯ρeiτcωox2)h1(−1) + ¯σ(ρx1 + ¯ρx2)h1(0)]
where
h1(−1) = h120(−1)x21 +h111(−1)x1x2+h102(−1)x2, h1(0) =h120(0)x21+h111(0)x1x2+h102(0)x22. and hence
1
2ProjS[(Dyf21)h](x,0,0) =
C2x21x2
C¯2x1x22
, where
C2 = ατci(2)
D¯ [e−iω0τ0βρh111(−1) +ρ¯σh111(0) +eiω0τ0βρh¯ 120(−1) + ¯ρ¯σh120(0)].
Here h20, h11 are determined by System (14) and System (15). After long but basic calculations, we obtain
h120(0) =
(2i(2)αρ2e−3τcω0( ¯D(−ie2iτcω0(2De3iτcω0σ)ω(−iq+ 2ω0) +kαβ(β+e2iτcω0σ)((−1 +e2iτcω0)β−2ieiτcω0ω0)) +eiτcω0(β+e2iτcω0σ)(−ieiτcω0kαβ+ie3iτcω0kαβ
+2βω0+ 2e2iτcω0(q+ 2iω0)ω0)¯ρ) +D(2eiτcω0ρ(β+e2iτcω0(q+ 2iω0))ω0
−ikαβ((−1 +e2iτcω0)β+ρ−2e2iτcω0ρ−2ie3iτcω0ω0))(β+eiτcω0σ))¯
/(ω0D(−iβ+e2iτcω0(−iq+ 2ω0))−α(βγ−ie2iτcω0(k−γ)(−iq+ 2ω0)) ¯D), h120(−1) =
2i(2)αρ2e−5τcω0
ω0|D|2 [i(−1 +e2iτcω0)(e2iτcω0(β+e2iτcω0σ) ¯Dρ¯+Dρ(β+e2iτcω0σ))¯ +( ¯D(−ie2iτcω0(2De2iτcω0ω0(−iq+ 2ω) +kαβ(β+e2iτcω0σ)((−1 +eiτcω0)β
−2ie2iτcω0ω0)) +eiτcω0(β+e2iτcω0)(−ieiτcω0kαβ+ie3iτcω0kαβ+ 2βω0
+2e2iτcω0(q+ 2iω0)ω0)¯ρ) +D(2e2iτcω0ρ(β+e2iτcω0(q+ 2iω0))ω0
−ikαβ((−1 +e2iτcω0)β+ρ−e2iτcω0 −2ie3iτcω0))(β+e2iτcω0σ))]¯
/(2ω0(−iβ+e2iτcω0(−iq+ 2ω))−α(βγ−ie2iτcω0(k−γ)(−iq+ 2ω0)))
and
h111(0) =
4i(2)|ρ|2e−iτcω0
(βγ+ (−k+γ)q)|D|2[eiτcω0D(¯ −Dq+kαβ(β+σ)(−1 +eiτcω0βτc) +(β+σ)(−q+β(−1 +eiτcω0kατc))¯ρ)−D(eiτcω0(kαβ−(β+q)ρ) +kαβ(−β+ρ)τc)(β+ ¯σ)],
h111(−1) 4i(2)|ρ|2e−iτcω0
|D|2 [−e−2iτcω0ατc((β+σ) ¯Dρ¯+De−2iτcω0ρ(β+ ¯σ))
− 1
βγ+ (−k+γ)q( ¯Deiτcω0(−Dq+kαβ(β+σ)(−1 +eiτcω0βτc)
−(β+σ)(−q+β(−1 +eiτcω0kατc))¯ρ)−D(eiτcω0(kαβ−(β+q)ρ) +kαβ(−β+ρ)τc)(β+ ¯σ))].
Collecting the results above, we obtain 1
3!g31(x,0, µ) =
b21x21x2
¯b21x1x22
+O(µ2|x|),
where b21 = a21+ 12(C1 +C2). Therefore, System (10) can be transformed into the following normal form:
x˙1 =iτcω0x1+a1µx1 +b21x21x2+ h.o.t.,
˙
x2 =−iτcω0x2+ ¯a1µx2+ ¯b21x1x22+ h.o.t., (16) wherea1is given in (13). Letx1 =w1+iw2, x2 =w1−iw2andw1 =rcosξ, w2 =rsinξ.
Then (16) can be further written as
r˙=aµr+br3 + h.o.t., ξ˙=τcω0+ h.o.t.,
where a = Re[a1] and b = Re[b21]. Hence the first Lyapunov coefficient is l1(µ) = b+O(µ), see [3, 13].
Theorem 3.1. Let a and b be given above.
(i) The bifurcating periodic solution is stable if b <0, and unstable if b >0;
(ii) The Hopf bifurcation is supercritical if ab <0, and subcritical if ab > 0.
Remark. The coefficient a is given by a= Re[a1] =
k2α2(q2 +ω02)
β2(α2γ2+ω20)|D|2[−αω02(β2−q2−ω02)(β2γ+ (k−γ)(q2+ω02)) +α3(−β4γ3+β2γ(k2−3kγ+ 2γ2)(q2+ω02) + (k−γ)3(q2+ω20)2 +ω04(β2q−(q2+ω02)(q−ω02))−α4(k−γ)2((k−γ)2q(1 +qτc)(q2+ω02)
−β2γ2(q+q2τc+τcω20)) +α2ω02(2kγ(−β2q+ 2q3+q4τc+ 2qω02−τcω40) +k2(β2q−2q3−q4τc−2qω2+τcω04) +γ2(−(q2+ω02)(2q+q2τc−τcω02) +β2(2q+q2τc +τcω02))))].
Although the explicit algorithm is derived to compute b, it is difficult to determine the sign of b for general α, β, γ, k, q. But if i(2) = 0, it is easy to see C1 = C2 = 0 and hence b can be simply expressed as
b = −3i(3)
|D|2(β2+q2+ω02+ 2βqcos(τcω0)−2βωsin(τcω0))(−kαβ2q+ 2β2q +2β2q2+q4−kαβ2q2τc+ 2β2ω20+ 2q2ω2−kα2β2τcω20+ω04
+β(β2q+ 3q(q2+ω02)−kα(q2+q3τc−ω02+qτcω20)) cos(τcω0) +β2(q2−ω20) cos(2τcω0)−β3ω0sin(τcω0) + 2kαβqsin(τcω0)
−3βq2ω0sin(τcω0)
+kαβq2τcω0sin(τcω0)−3βω03sin(τcω0)−2β3qω0sin(2τcω0)).
4 Numerical Simulations
In this section, we give some examples to illustrate the theoretical results obtained in the previous sections.
Example 1. Let α = 1, β = 0.8, γ = 0.5625, q = 0.9 and I(s) = tanh(0.5s).
Then (0,0) is an equilibrium point of System (2),k = 0.5, i(2) = 0, i(3) =−0.041667.
Hence ω+ = 0.6066, and τ0+ = 3.1382. Take τ = 2.5. According to Theorem 2.2 (ii), the trivial equilibrium point (0,0) is asymptotically stable, (Figure 1).
Example 2. Let α = 1, β = 0.8, γ = 0.5625, q = 0.9 and I(s) = tanh(0.5s).
0 10 20 30 40 50 60 70 80 90 100
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
t
Y(t)
0 10 20 30 40 50 60 70 80 90 100
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
t
K(t)
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Y
K
Figure 1: The equilibrium point (0,0) is asymptotically stable when τ < τ0+.
0 10 20 30 40 50 60 70 80 90 100
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
t
Y(t)
0 10 20 30 40 50 60 70 80 90 100
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
t
K(t)
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
−0.1
−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Y
K
Figure 2: The stable periodic orbit generated by Hopf bifurcation when β < q.
0 10 20 30 40 50 60
−0.02
−0.015
−0.01
−0.005 0 0.005 0.01 0.015
t
Y(t)
0 10 20 30 40 50 60
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15
t
K(t)
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
−0.1
−0.05 0 0.05 0.1
Y
K
Figure 3: The stable periodic orbit generated by Hopf bifurcation when β > q.